Subgroups — C₂×#405×C₂₇₇ #47980261

Order 35456.

592 conjugacy classes of subgroups.

The subgroup lattice has 152 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×M₁₆×C₂₇₇	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₈×C₂₇₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#117×C₂₇₇	C₂	1	1	yes	maximal, characteristic
4	#405×C₂₇₇	C₂	1	8	yes	maximal, direct product factor
5	C₂×#97×C₂₇₇	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×C₈×C₂₇₇	C₂	1	2	yes	maximal, complemented
7	C₂³×C₄×C₂₇₇	C₂²	1	1	yes	characteristic
8	C₂×M₁₆×C₂₇₇	C₄	1	8	yes	complemented
9	C₂×M₁₆×C₂₇₇	C₂²	1	4	yes	—
10	C₂²×C₈×C₂₇₇	C₄	1	8	yes	complemented
11	C₂²×C₈×C₂₇₇	C₂²	1	1	yes	characteristic
12	C₂²×C₈×C₂₇₇	C₂²	1	2	yes	—
13	C₂²×C₈×C₂₇₇	C₂²	1	4	yes	—
14	C₂²×C₈×C₂₇₇	C₂²	1	1	yes	characteristic
15	#117×C₂₇₇	C₄	1	4	yes	—
16	#117×C₂₇₇	C₂²	1	4	yes	—
17	C₂×C₄⋊C₄×C₂₇₇	C₄	1	2	yes	—
18	C₂×C₂²⋊C₄×C₂₇₇	C₄	1	2	yes	—
19	C₂×C₄²×C₂₇₇	C₂²	1	2	yes	—
20	#97×C₂₇₇	C₂²	1	8	yes	complemented
21	C₄×C₈×C₂₇₇	C₂²	1	8	yes	complemented
22	C₂⁴×C₂₇₇	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₂₇₇	C₂×C₄	1	4	yes	—
24	C₂²×C₄×C₂₇₇	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₂₇₇	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₂₇₇	C₂×C₄	1	4	yes	—
27	C₂²×C₄×C₂₇₇	C₂×C₄	1	4	yes	—
28	C₂²×C₄×C₂₇₇	C₂³	1	1	yes	characteristic
29	C₂²×C₄×C₂₇₇	C₂³	1	2	yes	—
30	M₁₆×C₂₇₇	C₂×C₄	1	16	yes	complemented
31	C₂×C₈×C₂₇₇	C₂×C₄	1	4	yes	—
32	C₂×C₈×C₂₇₇	C₂×C₄	1	4	yes	—
33	C₂×C₈×C₂₇₇	C₂×C₄	1	8	yes	—
34	C₂×C₈×C₂₇₇	C₂×C₄	1	16	yes	complemented
35	C₂×C₈×C₂₇₇	C₂³	1	2	yes	—
36	C₂×C₈×C₂₇₇	C₂³	1	4	yes	—
37	C₂×C₈×C₂₇₇	C₂³	1	2	yes	—
38	C₄⋊C₄×C₂₇₇	C₂×C₄	1	8	yes	—
39	C₂²⋊C₄×C₂₇₇	C₂×C₄	1	8	yes	—
40	C₄²×C₂₇₇	C₂×C₄	1	4	yes	—
41	C₄²×C₂₇₇	C₂³	1	4	yes	—
42	C₂³×C₂₇₇	—	2	4	no	—
43	C₂³×C₂₇₇	C₄²	1	4	yes	—
44	C₂³×C₂₇₇	C₂²×C₄	1	2	yes	—
45	C₂³×C₂₇₇	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₂₇₇	—	2	8	no	—
47	C₂×C₄×C₂₇₇	C₄²	1	8	yes	—
48	C₂×C₄×C₂₇₇	C₄²	1	4	yes	—
49	C₂×C₄×C₂₇₇	C₂²×C₄	1	8	yes	—
50	C₂×C₄×C₂₇₇	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₂₇₇	C₂²×C₄	1	2	yes	—
52	C₂×C₄×C₂₇₇	C₂²×C₄	1	2	yes	—
53	C₂×C₄×C₂₇₇	C₂²×C₄	1	2	yes	—
54	C₂×C₄×C₂₇₇	C₂²×C₄	1	4	yes	—
55	C₂×C₄×C₂₇₇	C₂²×C₄	1	4	yes	—
56	C₂×C₄×C₂₇₇	C₂⁴	1	1	yes	characteristic
57	C₈×C₂₇₇	C₂²×C₄	1	4	yes	—
58	C₈×C₂₇₇	C₂²×C₄	1	4	yes	—
59	C₈×C₂₇₇	C₂²×C₄	1	8	yes	—
60	C₂²×C₂₇₇	—	2	4	no	—
61	C₂²×C₂₇₇	—	2	8	no	—
62	C₂²×C₂₇₇	C₂×C₄²	1	4	yes	—
63	C₂²×C₂₇₇	C₂×C₄²	1	2	yes	—
64	C₂²×C₂₇₇	#131	1	4	yes	—
65	C₂²×C₂₇₇	C₂³×C₄	1	1	yes	characteristic
66	C₄×C₂₇₇	—	2	8	no	—
67	C₄×C₂₇₇	C₂×C₄²	1	2	yes	—
68	C₄×C₂₇₇	C₂×C₄²	1	4	yes	—
69	C₄×C₂₇₇	C₂³×C₄	1	1	yes	characteristic
70	C₄×C₂₇₇	C₂³×C₄	1	1	yes	characteristic
71	C₂×C₂₇₇	—	2	4	no	—
72	C₂×C₂₇₇	#405	1	4	yes	direct product factor
73	C₂×C₂₇₇	C₂²×C₄²	1	1	yes	characteristic
74	C₂×C₂₇₇	C₂×#131	1	2	yes	—
75	C₂₇₇	C₂×#405	1	1	yes	characteristic, direct product factor, Sylow
76	C₂×#405	C₂₇₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
77	C₂²×M₁₆	C₂×C₂₇₇	1	1	yes	characteristic
78	C₂³×C₈	C₂×C₂₇₇	1	1	yes	characteristic
79	C₂×#117	C₂×C₂₇₇	1	1	yes	characteristic
80	#405	C₂×C₂₇₇	1	8	yes	direct product factor
81	C₂×#97	C₂×C₂₇₇	1	2	yes	complemented
82	C₂×C₄×C₈	C₂×C₂₇₇	1	2	yes	complemented
83	C₂³×C₄	C₂²×C₂₇₇	1	1	yes	characteristic
84	C₂×M₁₆	C₄×C₂₇₇	1	8	yes	complemented
85	C₂×M₁₆	C₂²×C₂₇₇	1	4	yes	—
86	C₂²×C₈	C₄×C₂₇₇	1	8	yes	complemented
87	C₂²×C₈	C₂²×C₂₇₇	1	1	yes	characteristic
88	C₂²×C₈	C₂²×C₂₇₇	1	1	yes	characteristic
89	C₂²×C₈	C₂²×C₂₇₇	1	2	yes	—
90	C₂²×C₈	C₂²×C₂₇₇	1	4	yes	—
91	#117	C₄×C₂₇₇	1	4	yes	—
92	#117	C₂²×C₂₇₇	1	4	yes	—
93	C₂×C₄⋊C₄	C₄×C₂₇₇	1	2	yes	—
94	C₂×C₂²⋊C₄	C₄×C₂₇₇	1	2	yes	—
95	C₂×C₄²	C₂²×C₂₇₇	1	2	yes	—
96	#97	C₂²×C₂₇₇	1	8	yes	complemented
97	C₄×C₈	C₂²×C₂₇₇	1	8	yes	complemented
98	C₂⁴	C₂×C₄×C₂₇₇	1	1	yes	characteristic
99	C₂²×C₄	C₂×C₄×C₂₇₇	1	4	yes	—
100	C₂²×C₄	C₂×C₄×C₂₇₇	1	1	yes	characteristic
101	C₂²×C₄	C₂×C₄×C₂₇₇	1	2	yes	—
102	C₂²×C₄	C₂×C₄×C₂₇₇	1	4	yes	—
103	C₂²×C₄	C₂×C₄×C₂₇₇	1	4	yes	—
104	C₂²×C₄	C₂³×C₂₇₇	1	1	yes	characteristic
105	C₂²×C₄	C₂³×C₂₇₇	1	2	yes	—
106	M₁₆	C₂×C₄×C₂₇₇	1	16	yes	complemented
107	C₂×C₈	C₂×C₄×C₂₇₇	1	4	yes	—
108	C₂×C₈	C₂×C₄×C₂₇₇	1	4	yes	—
109	C₂×C₈	C₂×C₄×C₂₇₇	1	8	yes	—
110	C₂×C₈	C₂×C₄×C₂₇₇	1	16	yes	complemented
111	C₂×C₈	C₂³×C₂₇₇	1	2	yes	—
112	C₂×C₈	C₂³×C₂₇₇	1	4	yes	—
113	C₂×C₈	C₂³×C₂₇₇	1	2	yes	—
114	C₄⋊C₄	C₂×C₄×C₂₇₇	1	8	yes	—
115	C₂²⋊C₄	C₂×C₄×C₂₇₇	1	8	yes	—
116	C₄²	C₂×C₄×C₂₇₇	1	4	yes	—
117	C₄²	C₂³×C₂₇₇	1	4	yes	—
118	C₂³	—	2	4	no	—
119	C₂³	C₄²×C₂₇₇	1	4	yes	—
120	C₂³	C₂²×C₄×C₂₇₇	1	2	yes	—
121	C₂³	C₂²×C₄×C₂₇₇	1	1	yes	characteristic
122	C₂×C₄	—	2	8	no	—
123	C₂×C₄	C₄²×C₂₇₇	1	8	yes	—
124	C₂×C₄	C₄²×C₂₇₇	1	4	yes	—
125	C₂×C₄	C₂²×C₄×C₂₇₇	1	8	yes	—
126	C₂×C₄	C₂²×C₄×C₂₇₇	1	1	yes	characteristic
127	C₂×C₄	C₂²×C₄×C₂₇₇	1	2	yes	—
128	C₂×C₄	C₂²×C₄×C₂₇₇	1	2	yes	—
129	C₂×C₄	C₂²×C₄×C₂₇₇	1	2	yes	—
130	C₂×C₄	C₂²×C₄×C₂₇₇	1	4	yes	—
131	C₂×C₄	C₂²×C₄×C₂₇₇	1	4	yes	—
132	C₂×C₄	C₂⁴×C₂₇₇	1	1	yes	characteristic
133	C₈	C₂²×C₄×C₂₇₇	1	4	yes	—
134	C₈	C₂²×C₄×C₂₇₇	1	4	yes	—
135	C₈	C₂²×C₄×C₂₇₇	1	8	yes	—
136	C₂²	—	2	4	no	—
137	C₂²	—	2	8	no	—
138	C₂²	C₂×C₄²×C₂₇₇	1	4	yes	—
139	C₂²	C₂×C₄²×C₂₇₇	1	2	yes	—
140	C₂²	#131×C₂₇₇	1	4	yes	—
141	C₂²	C₂³×C₄×C₂₇₇	1	1	yes	characteristic
142	C₄	—	2	8	no	—
143	C₄	C₂×C₄²×C₂₇₇	1	2	yes	—
144	C₄	C₂×C₄²×C₂₇₇	1	4	yes	—
145	C₄	C₂³×C₄×C₂₇₇	1	1	yes	characteristic
146	C₄	C₂³×C₄×C₂₇₇	1	1	yes	characteristic
147	C₂	—	2	4	no	—
148	C₂	#405×C₂₇₇	1	4	yes	direct product factor
149	C₂	C₂²×C₄²×C₂₇₇	1	1	yes	characteristic
150	C₂	C₂×#131×C₂₇₇	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈×C₂₇₇	C₂²
commutator	C₂	C₂²×C₄²×C₂₇₇
Frattini, cosocle	C₂×C₄	C₂⁴×C₂₇₇
socle	C₂³×C₂₇₇	C₂²×C₄
prime radical	C₂⁴×C₂₇₇	C₂×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈×C₂₇₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
4. #405×C₂₇₇	C₂	direct (N × H)
5. C₂×#97×C₂₇₇	C₂	semidirect
6. C₂×C₄×C₈×C₂₇₇	C₂	semidirect
8. C₂×M₁₆×C₂₇₇	C₄	semidirect
10. C₂²×C₈×C₂₇₇	C₄	semidirect
20. #97×C₂₇₇	C₂²	semidirect
21. C₄×C₈×C₂₇₇	C₂²	semidirect
30. M₁₆×C₂₇₇	C₂×C₄	semidirect
34. C₂×C₈×C₂₇₇	C₂×C₄	semidirect
72. C₂×C₂₇₇	#405	direct (N × H)
75. C₂₇₇	C₂×#405	direct (N × H)
76. C₂×#405	C₂₇₇	direct (N × H)
80. #405	C₂×C₂₇₇	direct (N × H)
81. C₂×#97	C₂×C₂₇₇	semidirect
82. C₂×C₄×C₈	C₂×C₂₇₇	semidirect
84. C₂×M₁₆	C₄×C₂₇₇	semidirect
86. C₂²×C₈	C₄×C₂₇₇	semidirect
96. #97	C₂²×C₂₇₇	semidirect
97. C₄×C₈	C₂²×C₂₇₇	semidirect
106. M₁₆	C₂×C₄×C₂₇₇	semidirect
110. C₂×C₈	C₂×C₄×C₂₇₇	semidirect
148. C₂	#405×C₂₇₇	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×M₁₆×C₂₇₇	2
C₂³×C₈×C₂₇₇	2
C₂×#117×C₂₇₇	2
#405×C₂₇₇	2
C₂×#97×C₂₇₇	2
C₂×C₄×C₈×C₂₇₇	2
C₂×#405	277

Subgroups — C2×#405×C277 #47980261

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#405×C₂₇₇ #47980261